>>>>Hi all
>>>>
>>>>I'm having a mathematic difficulty. I've read all I can find on the subject. I need a formula. I have a circle with a hole in it, cut along a radius and pulled into a helix. I want a formula to return the intersections of this circle with an inner cylinder at 0, 90, 180 and 270 degrees and with an outer cylinder at 0, 60, 120, 180, 240 and 300 degrees. The radius of the inner and outer cylinders is known. The height the helix climbs along the inner and outer cylinders after one full turn is also known.
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>>>>So, I want x, y and z at each angle and radius, but I do not know the height (z) of the intersection points as they spiral through 3d space. I can obtain x and y on a 2D unit circle using sin, cos and tan (and probably an array), but can't determine z.
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>>>>Hope I explained that well.
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>>>If you know the height the helix climbs on the inside and outside cylinders, is there any reason the climb rate is not constant as you rotate i.e. linear rate of climb? In other words, if you rotate 180deg from the bottom, are you not exactly half way up both the inside and outside climbs?
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>>I'm looking at a physical model wherein that does not seem to be the case, because it is buckling out of shape. Your insight might remove the buckling.
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>>Thanks. I'll give it a try.
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>Roughly, what is the ratio of the final length of the helix compared to the diameter of the initial circle? What are the ratios of the initial inner and outer circles? What material is the physical model made of?
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>The higher the ratio of helix length over initial circle diameter, the greater the chance that highly-stressed parts of any physical material/model will be stretched past the material's elastic limit/yield point and will deform plastically (if not fail completely). Since plastic deformation after material yield is essentially unpredictable, it's not surprising a physical model may show unpredictable buckling.
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>Also, the higher the ratio of the initial circle diameter compared to the diameter of the initial hole, the higher the stresses. If the ratio is low (not much over 1) you end up with something that would look like a coil spring. Coil springs are familiar everyday items but they are not produced by your helix-production method, I believe wire is hot-wound around a mandrel and then heat-treated.
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>There's another thing to bear in mind if you're creating physical helices in the manner you describe. That is, how the material is gripped before being stretched. One way would be to clamp both inside edges horizontally, then stretch. However, if both inside edges are forced to remain horizontal then the slope of the helix near the top and bottom will change rapidly. This will cause large stresses in the material at top and bottom, and a non-linear slope that will end up looking like an integral sign. Ideally, the clamps should allow rotation (in 2 axes) of the material being held, while being stretched.
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>I guess the biggest question is, is this just a mathematically ideal problem, or are you trying to do something related to production of real helices in this fashion?
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>If ideal, you can assume a perfect material that does not suffer yield failure, and perfect gripping of the ends of the helix. With that, you can probably assume linear slope.
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>If you're working with real helices, the most important thing is to pick geometry and materials such that you don't suffer material yield. If the material doesn't yield, the final structure of physical models should be consistent. Determining the final shape mathematically depends on stress distribution, which for a helix like you propose will be very complex and will almost certainly require FEA. Unless you already happen to be expert in FEA I'm fairly certain you'd be better off measuring physical models.

Hi Al

It's for a physical model. It should undergo very little stress. I have been trying to make the shape by hand without any appreciation of the math, I think I was stretching the material and causing buckling.